Abstract

We introduce and study well-posedness in connection with the symmetric vector quasi-equilibrium problem, which unifies its Hadamard and Levitin-Polyak well-posedness. Using the nonlinear scalarization function, we give some sufficient conditions to guarantee the existence of well-posedness for the symmetric vector quasi-equilibrium problem.

1. Introduction

Vector equilibrium problem provides a very general model for many problems like the vector variational inequality problem, the vector complementarity problem, the vector optimization problem, the multiobjective game problem, the vector network equilibrium problem, and the vector saddle point problem (see, e.g., [1–3] and the references therein).

Well-posedness plays an important role in the theory and numerical methods for optimization. The first concept of well-posedness was introduced by Tykhonov [4] for a global minimization problem having a unique solution. Lucchetti and Patrone [5] introduced the notion of well-posedness for variational inequalities. Lignola and Morgan [6] studied the well-posedness of optimization problems with variational inequality constraints. Fang et al. [7] investigated well-posedness for equilibrium problems and optimization problems with equilibrium constraints. Hu et al. [8] studied well-posedness of systems of equilibrium problems.

The notions of well-posedness can be mainly divided into three groups, namely, Hadamard type, Tykhonov type, and Levitin-Polyak type. Researchers have studied the relations between the Hadamard well-posedness and Tykhonov well-posedness for different problems (see [9, 10]). Most of the literature deals with directly specific notions of well-posedness. Huang et al. [11] investigated the Levitin-Polyak well-posedness of variational inequalities problems with functional constraints. S. J. Li and M. H. Li [12] studied the Levitin-Polyak well-posedness of vector equilibrium problems. Li et al. [13] investigated Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. Peng et al. [14] studied Levitin-Polyak well-posedness of generalized vector equilibrium problems with both abstract set constraints and functional constraints. Salamon [15] considered the Hadamard well-posedness by using the vector topological pseudomonotonicity. Peng et al. [16] investigated the Hadamard well-posedness of vector equilibrium problems by considering the perturbations of both vector-valued functions and feasible sets. Li and Zhang [17] studied the Hadamard well-posedness for the vector-valued optimization problems. Long and Huang [18] considered the -well posedness for the symmetric quasi-equilibrium problems. Recently, Han and Gong [19] studied the generalized Levitin-Polyak well-posedness of symmetric strong vector quasi-equilibrium problems. Deng and Xiang [20] introduced and studied the generalized well-posedness for the generalized vector equilibrium, which unifies its Hadamard and Levitin-Polyak well-posedness.

Motivated and inspired by the papers mentioned above, in this paper, we introduce a well-posedness concept for the symmetric vector quasi-equilibrium problem, which unifies its Hadamard and Levitin-polyak well-posedness. By employing the scalarization function, we give some sufficient conditions to guarantee the existence of the well-posedness for the symmetric vector quasi-equilibrium problem in real locally convex Hausdorff topological vector spaces. The results presented in this paper generalize and extend Theorem 4.2 of [16] and Theorem 4.1 of [20].

2. Preliminaries

Let and be nonempty subsets of real locally convex Hausdorff topological vector spaces and , respectively. Assume and are two set-valued mappings. Let be a real topological vector space and be a closed convex pointed cone with , where denotes the topological interior of . It is well known that the cone can induce the following orders:

Let and be two vector valued functions. Let and be two closed convex pointed cones of with and . In this paper, we consider the following the symmetric vector quasi-equilibrium problem (in short SVQEP): find such that , , and

Some special cases of SVQEP are as follows.

(I) If , , and for all , where are two mappings, then (SVQEP) reduces to the problem of finding such that , , and
This problem was studied in Fu [21] and Han and Gong [19].

(II) If and , then (SVQEP) reduces to the following problem: find such that , , and

(III) If , , , and for all , where are two mappings, then (SVQEP) reduces to the symmetric quasi-equilibrium problem: find such that , , and
This problem was considered in Long and Huang [18].

It is well known that SVQEP includes many important problems as special cases, such as equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed point problems.

Now we recall some useful definitions and lemmas.

Let be a metric space. Denote a family of all nonempty compact subsets of by . For any , let
denote the Hausdorff metric on . It is well known that is complete if and only if is complete.

Definition 1. A sequence is called a Levitin-Polyak (in short LP) approximating solution sequence converging to for SVQEP, if there exists with such that
where and are given points.

Definition 2 (see [3]). Let , be two real Hausdorff topological spaces, a nonempty subset, and a closed convex pointed cone. A mapping is said to be(i)upper semicontinuous (-u.s.c.) (resp., -lower semicontinuous (-l.s.c.) at if, for any neighborhood of zero in , there exists a neighborhood of zero in such that
(ii)-u.s.c. (resp., -l.s.c.) on if it is -u.s.c. (resp., -l.s.c.) at every point .

Definition 3 (see [22]). Let and be two topological spaces. A set-valued mapping is said to be(i)upper semicontinuous (u.s.c.) at if, for any open set , there is an open neighborhood of such that for each ,(ii)lower semicontinuous (l.s.c.) at if, for any open set , there is an open neighborhood of such that for each ,(iii)continuous at if it is both upper and lower semicontinuous at ,(iv)upper semicontinuous (lower semicontinuous or continuous) on if it is upper semicontinuous (lower semicontinuous or continuous) at every ,(v)an usco mapping if is upper semicontinuous on and is compact for each ,(vi)closed if and only if its graph is closed.

Lemma 4 (see [23]). Let and be two topological spaces and a set-valued mapping. Suppose that is compact. Then is closed if and only if it is upper semicontinuous.

Lemma 5 (see [24]). Let and all be nonempty compact subsets of a metric space with in the Hausdorff metric topology. Then the following statements hold.(i) is also nonempty compact subset of .(ii)If , then .(iii)For any , there exists such that .

Lemma 6 (see [25]). Let and be two topological spaces and be an usco mapping. Then for any net with and , there exists a subnet such that .

Lemma 7 (see [26]). For any fixed point , the nonlinear scalarization function is defined by
The nonlinear scalarization function has the following properties:(i),(ii),(iii),(iv),(v), where denotes the topological boundary of .

Lemma 8. If is -upper semicontinuous, then is upper semicontinuous.

Proof. For any fixed , let
In order to show that is upper semicontinuous, we only need to show that is closed. Letting with , then
and Lemma 7 shows that
Suppose to the contrary that . We have
It follows from Lemma 7 that
This implies that there exists a neighborhood of zero element in such that
Since is -upper semicontinuous, for the above , there exists a positive integral number such that, when , one has
From (15) and (16), when , we have
which is in contradiction with (12). Thus, and so is upper semicontinuous. This completes the proof.

Example 9. Let and . Let
Then it is easy to check that is -upper semicontinuous and so Lemma 8 shows that is upper semicontinuous for any .

3. Bounded Rationality Model and Definition of Well-Posedness for SVQEP

Let and be two metric spaces and be a Banach space with a norm . Let and be nonempty closed convex pointed cones of with apex at the origin such that and .

In this section, we first define the problem space of SVQEP as follows: is upper semicontinuous on is upper semicontinuous on and for all , , and are continuous with compact values, and there exists such that and for all and .

For any with , define
where and denote the Hausdorff metric on and , respectively. Then it is easy to see that is a metric space.

Next we define the bounded rationality model for (SVQEP) as follows.(i), and are metric spaces.(ii)The feasible set of the symmetric vector quasi-equilibrium problems is defined by
(iii)The solution set of problem is defined by
(iv)For any , define
(v)The rationality function of the problem is defined by
where and .

Lemma 10. (i) For any and , .(ii) For any , .(iii) For and with , if and only if
In particular, if and only if and .

Proof. (i) For any and , we have and . From Lemma 7(v), we know that and , since and . Thus, one has
(ii) It is obvious that for all .(iii) Assume that and such that . Then we have
and so
with
It follows that
By Lemma 7(iii), one has
Conversely, assume that and with such that
Then Lemma 7(iii) shows that
These inequalities imply that
Therefore, we get
This completes the proof.

Remark 11. Lemma 10 is a generalization of Lemma 3.1 of Deng and Xiang [20].

Example 12. Let , , and . Assume that
Then it is easy to see that
(i)If , then and . It is obvious that
(ii)For any , one has
(iii)It is easy to check that if and only if . Moreover, taking , then and .

By Definition 1 and Lemma 10, for all with , the LP approximating solution set for the problem is defined as
and the set of solution for the problem is defined as

Next we define the Levitin-Polyak well-posedness and the Hadamard well-posedness for (SVQEP) as follows.

Definition 13. (i) If, for any with , there exists a subsequence such that , then the problem is said to be generalized LP well-posedness.(ii) If (a singleton), for any with , there exists a subsequence such that , then the problem is said to be LP well-posedness.

Definition 14. (i) If, for any with and any , there exists a subsequence , such that , then the problem is said to be generalized Hadamard well-posed.(ii) If (a singleton), for any with and any , we have , then the problem is said to be Hadamard well-posed.

By Definitions 13 and 14, we introduce the definition of (generalized) well-posedness, which unifies its Levitin-Polyak well-posedness and Hadamard well-posedness.

Definition 15. (i) If for any with , , with , there exists a subsequence such that , then the problem is said to be generalized well-posed.(ii) If (a singleton), for any with , , with , we have , then the problem is said to be well-posed.

Lemma 16. If the problem is (generalized) well-posed, then(i)the problem must be (generalized) LP well-posed;(ii)the problem must be (generalized) Hadamard well-posed.

Proof. We only prove the statement of generalized well-posedness. The proof of the well-posedness is similar to the generalized well-posedness.(i) For any , letting , we know that and . Since the problem is generalized well-posed, there exists a subsequence such that . Thus, it follows that is generalized LP well-posed.(ii) For any with and , letting , it is easy to see that . Since the problem is generalized well-posed, there exists a subsequence such that . Therefore, we know that is generalized Hadamard well-posed.This completes the proof.

4. Some Sufficient Conditions for Well-Posedness of SVQEP

Assume and are compact metric spaces. In this section, we give some sufficient conditions to guarantee the existence of the well-posedness for SVQEP.

Lemma 17. is a complete metric space.

Proof. Let be a Cauchy sequence in . Then for any given , when integral numbers and are large enough, we have
It follows that and are both Cauchy sequences in for any fixed and , respectively. Moreover, we know that , are both Cauchy sequences in and for any fixed , respectively.(I) Since is a complete space, there exists such that . Let . We show that is -upper semicontinuous and
For any open neighborhood of zero in , since is -upper semicontinuous, there exists an open neighborhood of such that
Since , for large enough, one has
It follows from (43) and (44) that
and so is -upper semicontinuous.On the other hand, shows that
Since , it is easy to see that
Moreover, from the fact that and , we know that .(II) Similar to the proof of (I), we know that there exists a -upper semicontinuous mapping such that with and
(III) Since , are both Cauchy sequences, and are complete, we know that there exist sets and such that
Let and . Then the fact that and are both continuous with compact set-values shows that and are both continuous with compact set-values.(IV) We prove that there exist and such that
In fact, since , there exist sequences and such that , , and
with
Since and are compact, there exist subsequences and such that and . From the continuities of and , we have
When are large enough, one has
and so
holds for large enough. By the definition of the Hausdorff metric, we have
This together with (55) shows that and so . Similarly, we can get .Next we prove that
By contradiction, we assume that there exists such that
Then there exists an open neighborhood of zero in such that
By Lemma 5, there is a sequence with such that . Since is -upper semicontinuous and , we have
From (59) and (60), we get
which is in contradiction with the fact that for all .Similarly, we can show that satisfies
(V) Let . Then and . Therefore, is a complete metric space. This completes the proof.

Lemma 18. is an mapping.

Proof. It is easy to see that is closed for any given . In fact, for any
with and , we have and and so
It follows from the continuity of that and so . Similarly, we can get . Thus, and so is closed. Since is compact, we know that is compact for any given . In order to show that is an u.s.c. mapping, from Lemma 4, it is sufficient to show that is closed.Let with . Then the completeness of shows that . Moreover, we have and and
By the definition of the Hausdorff metric, it follows that
From (66), we have . This completes the proof.

Lemma 19. For any , the rationality function is lower semicontinuous at .

Proof. For any given , let
In order to show is lower semicontinuous, we only need to show that is closed. Let with . We show that , that is, , which is equivalent to
where and , that is,
which is equivalent to (by Lemma 7)
By way of contradiction, assume there exists or such that
or
Without loss of generality, assume that (71) holds. Then there exists a neighborhood of zero in such that
For , since and are compact with , by Lemma 5, there exists with . Since is -upper semicontinuous and , we have
It follows from (73) and (74) that
On the other hand, since , we know that and so
It follows that
Now Lemma 10 implies that
which is in contradiction with (75). Thus, we know that is lower semicontinuous. This completes the proof.

Theorem 20. For all , the problem is generalized well-posed. Moreover, for all , if (a singleton), then the problem is well-posed.

Proof. Let with and with . Then
By (79), there exists such that . Since is an usco mapping, by Lemma 6, there exists a subsequence such that . It follows that
and so
On the other hand, the lower semicontinuity of shows that
which implies that . It follows from (82) and (83) that and so the problem is generalized well-posed.Moreover, we show that, if , then the problem is well-posed. If the sequence does not converge to , then there exist an open neighborhood of and a subsequence of such that . By the proof of the first part, we know that . This is in contradiction with . This completes the proof.

Example 21. Let , , , and . Let
Then it is easy to see that and
with
Furthermore, by Theorem 20, the problem is generalized well-posedness.

Remark 22. When and , the well-posedness for the problem was studied by [20]. Theorem 20 presented in this paper can be considered as a generalization and extension of Theorem 4.1 in [20].

Corollary 23. For all , the problem is (generalized) Hadamard well-posed.

Corollary 24. For all , the problem is (generalized) LP well-posed.

Remark 25. When , , the generalized Hadamard well-posedness for the problem was studied by [16]. Corollary 23 presented in this paper generalizes and extends Theorem 4.2 in [16].

Remark 26. We note that, when
the generalized LP well-posedness for the symmetric vector quasi-equilibrium problem was studied by [19].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful comments and suggestions, which have improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (11171237).